Let \(u \in ACL(\Omega )\) .
Absolute continuity implies that the classical partial derivative \(\partial _k u\)  of \(u \in ACL(\Omega )\)  exist for every \(x \in \Omega \)  such that \(\tilde{x}_k \in E_k\) .
Since \(|\widetilde{\Omega }_k\setminus E_k|_{n-1} = 0\) , it follows by Fubini's theorem that \(\partial _k u\)  exists for almost every \(x \in \Omega \) .
Another application of Fubini's theorem shows that the classical partial derivative is equal to the weak partial derivative, see [1]}.
Since the partial derivatives exist almost everywhere, it follows that the gradient \(\nabla u\)  exists almost everywhere.
A function \(u \in ACL(\Omega )\)  is said to belong to \(ACL^\varphi (\Omega )\) , if \(|\nabla u| \in L^\varphi (\Omega )\) .
